Question: If $a$ is a constant such that $9x^2 + 24x + a$ is the square of a binomial, then what is $a$?
If $9x^2 +24x + a$ is the square of a binomial, then the binomial has the form $3x +b$ for some number $b$, because $(3x)^2 = 9x^2$.  So, we compare $(3x+b)^2$ to $9x^2 + 24x + a$. Expanding $(3x+b)^2$ gives \[(3x+b)^2 = (3x)^2 + 2(3x)(b) + b^2 = 9x^2 + 6bx + b^2.\]Equating the linear term of this to the linear term of $9x^2+24x+a$, we have $6bx=24x$, so $b=4$.  Equating the constant term of $9x^2 + 6bx + b^2$ to that of $9x^2 + 24x+a$ gives us $a=b^2 = \boxed{16}$.